The energy dependence of the spectral fluctuations in the interacting boson model (IBM) and its connections to the mean-field structures have been analyzed through adopting two statistical measures, the nearest neighbor level spacing distribution $P(S)$ measuring the chaoticity (regularity) in energy spectra and the $\Delta_3(L)$ statistics of Dyson and Metha measuring the spectral rigidity. Specifically, the statistical results as functions of the energy cutoff have been worked out for different dynamical situations including the U(5)-SU(3) and SU(3)-O(6) transitions as well as those near the AW arc of regularity. It is found that most of the changes in spectral fluctuations are triggered near the stationary points of the classical potential especially for the cases in the deformed region of the IBM phase diagram. The results thus justify the stationary point effects from the point of view of statistics. In addition, the approximate degeneracies in the $2^+$ spectrum on the AW arc is also revealed from the statistical calculations.

A scheme of solving the proton-neutron interacting boson model (IBM-2) in terms of the SU(3) basis is introduced, by which the IBM-2 coupled with an octupole boson is applied to describe the low-energy structure of the critical point nucleus, $^{152}$Sm. The results indicate that the spectral properties of both the positive-parity bands and negative-parity bands in this nucleus can be well captured by the IBM-2 calculations through a simple Hamiltonian, providing an example of the IBM-2 in a unified description of quadrupole and octupole states in transitional systems. In addition, a statistical analysis of the low-spin states in the model is also provided.

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